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The nonexistence of expansive homeomorphisms of dendroids. (English) Zbl 0707.54028

Summary: Let X be a compact metric space with metric d. A homeomorphism f of X is expansive if there exists \(c>0\) (called an expansive constant for f) such that \(d(f^ n(x),f^ n(y))\leq c\) for all integers n implies \(x=y\). This property is important in the topological theory of dynamical systems. It is known that the Cantor set, the 2-adic solenoid and the 2-torus admit expansive homeomorphisms. But there exists no expansive homeomorphism on an arc and a circle. Also a Peano continuum which contains a free arc or a 1-dimensional open ANR does not admit an expansive homeomorphism. In particular, 1-dimensional compact ANRs admit no expansive homeomorphism. The following problem arises: Is it true that no tree-like continuum admits an expansive homeomorphism? The author proves that no dendroid \((=arcwise\) connected tree-like continuum) admits an expansive homeomorphism, and no uniformly arcwise connected continuum admits an expansive homeomorphism.

MSC:

54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
54H20 Topological dynamics (MSC2010)
54E40 Special maps on metric spaces
37D99 Dynamical systems with hyperbolic behavior
54F15 Continua and generalizations
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